Aspects of Riemannian and Spin Geometry

Abstract

Differential geometry has a long history in mathematics, with much development being made in recent decades. In this thesis, we will only survey two branches of the field, namely Riemannian and spin geometry.

We first deal with the preliminaries. We provide a brief discussion on the important concepts of differential geometry: differentiable manifolds, vector fields, differential forms, exterior differentiation, differential and pullbacks, tensor algebras and fields. We also give a summary for Lie groups and algebras, and for principal fibre bundles and associated fibre bundles.

We then study the theory of connections on a principal fibre bundle. We will introduce fundamental vector fields, connection forms and curvature forms. The essential idea behind connections is to introduce parallelism to manifolds. As such, we can define parallel displacement of fibres. We can extend the same idea to fibre bundles in general. In particular, we introduce covariant differentiation for vector bundles. We will focus on the bundle of linear frames, a principal fibre bundle associated to the tangent bundle of a manifold. We will define curvature and torsion tensors from this framework and observe how they will lead to the Bianchi identities. With all these concepts in place, we are capable to introduce Riemannian geometry. Finally, we provide an alternative definition of covariant differentiation, which is more useful in introducing spin geometry, the second focus of the thesis.

Before introducing the spin geometry, we study Clifford algebras and spin groups, and in particular, the Clifford algebras over Rn and its complexified counterpart. We then move on to discuss Clifford and spinor bundles, followed by connections on spinor bundles. This task is made easily after our lengthy discussion of the theory of connections made earlier. We then focus on spinor bundles created from oriented Riemannian manifolds. Finally, we introduce Dirac operators before finishing with Bochner-type identities.

In the last chapter, I will attempt to give an overview on some of the uses of the machinery discussed so far in Riemannian and spin geometry.